Spending Strategies differ from ship to ship, and from player to player. You have to balance what you spend on upgrades and what you spend at stores/special events. No strategy is perfect because the game has chance-based elements.
Shields should be a high priority, especially level 2, as they are fairly cheap and make you nigh-invincible for a few sectors, saving you more scrap than you spend. Similarly, engines up to level 3 are hilariously cheap and can give you a little luck-based defense, while defense drone mark 1 (especially with an existing drone system) helps immensely with the unavoidable rockets and irritating asteroids.
One upgrade that is advisable for blue events is level 2 Medbay, as it prevents you from losing a crew member from some kind of events and increases the chances of getting a new crew member from others. Furthermore, it can lead to a decent payout if you stumble upon the Unknown Disease on Mining Colony distress beacon. Some players make this their first purchase. Another early purchase is lvl 2 of the auto pilot for two reasons. One, you lose your entire dodge bonus and your jump engine stops charging if you suffer any damage to pilot system lvl 1. Two, the event, Plasma Storm Incapacitated Ships, can cost you a crewmember if you don't have a cloning bay. It should be noted that the plasma storm event seems to occur less often in the AE.
A reliable method of killing enemy crew - either boarding, fires, hacking drone to the medbay or O2, or meticulous use of an Anti-Bio Beam - is also a good idea, since it significantly increases scrap, resource, and crew gain.
Scrap Recovery Arms are always excellent investments (especially in the early-mid game, after the above is accomplished), as they can return their cost in sector 7 alone. Similarly, Drone Recovery Arms (especially with Hull Repair drones) can save quite a bit of money.
One savings plan is to keep about 150 scrap in the bank. This amount allows you to buy an expensive item (shield, cloak, or Weapon Pre-Igniter), or an array of medium purchases (Drone Control, Teleporter, Burst Laser Mk II, Rockman, etc) should you find a good store. It also allows you to make emergency upgrades and repairs if you stumble upon a tough encounter.
Generally, you should spend almost all your scrap on things that will return the scrap invested in the long run (see Early Priorities) in the first few sectors. Later, as improving your arsenal becomes priority, you should save more. A good way to combine these two goals is to keep very little on hand (maybe 40 scrap), but keep situational equipment, and upon seeing a store (or even better, a pair of stores), circle around and collect scrap before checking them out (and selling weapons/augments you don't really need).